Dissection puzzle

ABSTRACT

A dissection puzzle comprising a set of pieces, each piece in the set having an obverse face and a reverse face and a plurality of straight edges about said faces, and wherein said set of pieces may be formed into an assembled puzzle of straight-sided geometric shape by arranging the pieces in edge-to-edge abutting relationship, the form of the pieces being selected such that the area of each piece is selected from a set areas of different rational ratios to the area of the assembled puzzle and characterised in that: the obverse face and the reverse face of each piece include markings selected from a predetermined set of markings such that when arranged to form the assembled puzzle and viewed from one side, each edge of each piece adjacent another piece is adjacent a piece having a different marking.

FIELD OF INVENTION

This invention relates to a dissection puzzle. The present invention isprimarily directed to dissection puzzle using what has become known byvarious names including the “ostomachion” and “loculus of Archimedes”.The general area of the art is sometimes known as recreationalmathematics. However, the invention is limited to neither theostomachion as such nor to such field of use.

BACKGROUND ART

Teaching mathematics, particularly geometry, is sometimes perceived asbeing difficult on the presumption that the subject is essentiallyuninteresting by its nature. In short, the subject as normally taught isprosaic in both normal meanings of the word. Puzzles have been proposedto encourage students to learn arithmetic by turning learning time intoplaytime with little or no compromise on the learning component of theexperience. Dissection puzzles have been proposed to assist with theeducation of students of mathematics. Such puzzles fall into thecategory of recreational mathematics which includes such well knownpuzzles as the Chinese tangram, the term “tangram” itself sometimesbeing used to refer to dissection puzzles generally. In thisspecification, unless the context requires otherwise, the term“dissection puzzle” refers to puzzles of the type where a number ofstraight-edged pieces may be placed in edge-to-edge abuttingrelationship to form a predetermined straight-sided geometric shape.Bearing the above in mind, it might be useful to extend the utility ofdissection puzzles involving shaped pieces to include an arithmeticdimension.

In this specification, unless the context requires otherwise, the termloculus of Archimedes may be taken to mean a dissection puzzle wherein asquare is divided into fourteen straight-sided pieces, each piececircumscribing an area selected from a set areas of different rationalratios to the area of the square, although it will be appreciated thatit is not necessarily common general knowledge in the art to refer tothe loculus of Archimedes as a dissection puzzle.

The present invention aims to provide a dissection puzzle whichalleviates one or more of the aforementioned deficiencies of the priorart. Another aim is to provide a dissection puzzle which aid inenlivening interest in learning mathematics. Other aims and advantagesof the invention may become apparent from the following description.

DISCLOSURE OF THE INVENTION

With the foregoing in view, the present invention in one aspect residesbroadly in a dissection puzzle comprising a set of pieces, each piece inthe set having an obverse face and a reverse face and a plurality ofstraight edges about said faces, and wherein said set of pieces may beformed into an assembled puzzle of straight-sided geometric shape byarranging the pieces in edge-to-edge abutting relationship, the form ofthe pieces being selected such that the area of each piece is selectedfrom a set areas of different rational ratios to the area of theassembled puzzle and characterised in that:

the obverse face and the reverse face of each piece include markingsselected from a predetermined set of markings such that when arranged toform the assembled puzzle and viewed from one side, each edge of eachpiece adjacent another piece is adjacent a piece having a differentmarking.

Preferably, the markings are also arranged such that when arranged toform the assembled puzzle, there is provided an alternative arrangementof pieces such that the assembled puzzle is comprised of a plurality ofregions, one for each marking and of equal area to each of the otherregions. The rational ratios may also be selected such that one or moreof the areas are in rational ratio to one or more of the other areas.

In another aspect, the present invention resides broadly in a dissectionpuzzle comprising a set of pieces, each piece in the set having anobverse face and a reverse face and a plurality of straight edges aboutsaid faces, and wherein said set of pieces may be formed into anassembled puzzle of straight-sided geometric shape by arranging thepieces in edge-to-edge abutting relationship, the form of the piecesbeing selected such that the area of each piece is selected from a setareas of different rational ratios to the area of the assembled puzzleand characterised in that:

the obverse face and the reverse face of each piece include markingsselected from a predetermined set of markings such that when arranged toform the assembled puzzle and viewed from one side, each edge of eachpiece adjacent another piece is adjacent a piece having a differentmarking, and

the markings are also arranged such that when arranged to form theassembled puzzle, there is provided an alternative arrangement of piecessuch that the assembled puzzle is comprised of a plurality of regions,one for each marking and of equal area to each of the other regions.

In another aspect, the present invention in one aspect resides broadlyin a dissection puzzle comprising a set of pieces, each playing piece inthe set having an obverse face and a reverse face and a plurality ofstraight edges about said faces, and wherein said set of pieces may beformed into an assembled puzzle in the form of a straight-sidedgeometric shape by assembling the pieces in edge-to-edge abuttingrelationship, the form of the pieces being selected such that the areaof each piece is selected from a set areas of different rational ratiosto the area of the assembled puzzle and characterised in that:

the obverse face and the reverse face of each piece include markingsselected from a predetermined set of markings such that the pieces maybe assembled to form an assembled puzzle comprising contiguous areas foreach marking in exact ratio for the number of different markings.

In another aspect, the present invention reside broadly in a dissectionpuzzle comprising a set of pieces, each playing piece in the set havingan obverse face and a reverse face and a plurality of straight edgesabout said faces, and wherein said set of pieces may be formed into anassembled puzzle in the form of a straight-sided geometric shape byassembling the pieces in edge-to-edge abutting relationship, the form ofthe pieces being selected such that the area of each piece is selectedfrom a set areas of different rational ratios to the area of theassembled puzzle and characterised in that:

the obverse face and the reverse face of each piece include markingsselected from a predetermined set of markings such that said assembledpuzzle may be formed from all of the pieces with all but one of themarkings common to one side of the assembled puzzle.

Preferably, the assembled puzzle is square in shape and the set ofpieces is comprised of fourteen pieces. In such form, the piecespreferably conform to the shapes comprising the loculus of Archimedes.It is preferred that the markings be provided in the form of distinctivecolours. It is preferred that the predetermined set of marking comprisesfour different markings. For example, the three primary colours andblack may be selected for the colour markings on the faces of thepieces.

In another aspect, the present invention resides broadly in a dissectionpuzzle comprising a set of pieces, each piece in the set having anobverse face and a reverse face and a plurality of straight edges aboutsaid faces, and wherein said set of pieces may be formed into anassembled puzzle of straight-sided geometric shape by arranging thepieces in edge-to-edge abutting relationship, the form of the piecesbeing selected such that the area of each piece being a rational ratioto the area of the assembled puzzle, and the area of at least some ofthe pieces being in rational ratio to the other pieces, andcharacterised in that:

the obverse face and the reverse face of each piece include markingsselected from a predetermined set of markings such that when arranged toform the assembled puzzle and viewed from one side, each edge of eachpiece adjacent another piece is adjacent a piece having a differentmarking.

In another aspect, the present invention resides broadly in a dissectionpuzzle comprising a set of pieces conforming to the loculus ofArchimedes, each piece in the set having an obverse face and a reverseface and a plurality of straight edges about said faces, and whereinsaid set of pieces may be formed into an assembled puzzle ofstraight-sided geometric shape by arranging the pieces in edge-to-edgeabutting relationship and characterised in that:

the obverse face and the reverse face of each piece include markingsselected from a predetermined set of markings such that when arranged toform the assembled puzzle and viewed from one side, each edge of eachpiece adjacent another piece is adjacent a piece having a differentmarking.

BRIEF DESCRIPTION OF THE DRAWINGS

In order that the invention may be more readily understood and put intopractical effect, one preferred embodiment of the present invention willdescribed with reference to the following drawings, and wherein:

FIG. 1 is a diagrammatic representation of the dissection puzzle in theform of a solved loculus of Archimedes known in the art;

FIG. 2 shows the set of pieces comprising a preferred embodiment of thedissection puzzle according to the invention;

FIG. 3 shows the obverse faces of the pieces of the set of FIG. 2;

FIG. 4 shows the reverse faces of the pieces of FIG. 3; and

FIGS. 5 and 6 each show one arrangement of pieces of the puzzle of FIGS.1 to 4 into a square in accordance with the invention.

DETAILED DESCRIPTION OF THE DRAWINGS

The loculus of Archimedes 20 shown in FIG. 1 comprises the fourteenpieces numbered 1 to 14 against a twelve-by-twelve square grid todemonstrate the feature that each piece is in rational ratio in area tothe square. The pieces all fit in to a large square 21, the dimensionsof the large square being indicated by the 144 small squares indicatedtypically by reference numeral 22. In FIGS. 2 to 4, each piece of thepuzzle is shown with the same piece number as is given in FIG. 1.However, the pieces are coloured on the obverse and reverse faces toprovide the features of the invention herein defined and described. Theallocation of the colours is set forth in Table 1.

The pieces 1 to 7 and 9 to 12 are all scalene triangles. Piece 8 and 14are quadrilaterals and piece 13 is a pentagon. The pieces 1 and 7 areright triangles, and the quadrilateral 14 includes a right angle. Thepentagon is an uneven sided pentagon, but has two parallel sides and tworight angles. Although there are many solutions for arranging the piecesinto a square, the obverse and reverse sides are selected for thesolution as shown in FIG. 6 which has four equal and contiguous areas ofeach colour in the shape of a right triangle in which one of the sidesopposite the hypotenuse is double the length of the other.

In an educational setting, the 14 pieces may be used as an introductionto conservation and calculation of area, transformations, shapes andforms, logical deductive thinking, geometry of angles, combinations andpermutations, topology and possible many other areas of mathematics,particularly the mathematics involved in geometry. It may be observedthat pieces 9 and 10 are identical in shape, but exhibit the property ofchirality when one of the pieces is reversed. The puzzle may be solvedso that each puzzle piece is adjacent a piece of a different colour, andsuch an arrangement of the pieces may admit that pieces 9 and 10 andpieces 4 and 5 which are the chiral pieces of the set arranged with theopposite handed shape of each pair made visible. It is believed by theinventor that such an arrangement, absent, of course, the coloursselected to provide the present invention, may be the preferred solutionproposed by Archimedes, many solutions proposed by Archimedes being lostin antiquity. One such solution in accordance with the present inventionis shown in FIG. 5.

TABLE 1 Piece number Obverse Face Colour Reverse Face Colour 1 BlackYellow 2 Blue Black 3 Blue Yellow 4 Blue Red 5 Black Blue 6 Black Red 7Yellow Blue 8 Red Black 9 Black Yellow 10 Red Black 11 Red Blue 12 RedYellow 13 Blue Red 14 Yellow Red

It will be appreciated that the pieces forming the loculus of Archimedeshave corners which overlie an intersection of the grid lines of thetwelve-by-twelve grid in over which the pieces may be placed to solvethe loculus of Archimedes puzzle. It has been suggested that there are576 different arrangements of the puzzle pieces into the square, but itis to be appreciated that the loculus of Archimedes was not developed toprovide a challenge to assemble the pieces into the square, nor todetermine the number of solutions to do so, but as a result of thechallenge to divide the square up into fourteen pieces each of rationalratio to the area of the square. In other words, the divisor selectedfor obtaining the number of pieces was a divisor in respect of which asimple even division into like pieces is not possible.

Insofar as use of the puzzle in the teaching of geometry of angles isconcerned, it has been discovered that propositions set forth inEuclid's “Elements Book 1” may be demonstrated. Pieces 9 and 10 can beused to demonstrate proposition 6 from Euclid's Elements Book 1. If in atriangle to angles be equal to one another, the sides which subtend theequal angles will also be equal to one another. In similar fashion, twoor more pieces of the puzzle of the present invention may be laid out ona flat surface to demonstrate at least propositions 8, 10, 11, 17, 18,19, 20, 21, 32, 33, 34, 37, 38, 41 and 46 from Euclid's “Elements Book1”. Propositions 11 to 14 from Book 1 by Archimedes on the “Equilibriumof Planes” can also be demonstrated. His proposition 14 says that itfollows at once from the previous proposition that the centre of gravityof any triangle is the intersection of the lines drawn from any twoangles to the middle points of the opposite sides respectively. This canbe demonstrated by pieces 10, 11 and 12, the longest side of piece 12placed against the second longest side of piece 11 whereupon theshortest sides of each triangle in alignment. Piece 10 may then beplaced so that one of its sides matches the sum of the lengths of thetwo shortest sides of pieces 11 and 12, whereupon one of the other sidesof the triangle of piece 10 is in alignment with the second longest sideof piece 12.

Different pieces may also be laid against one another or in combinationto produce parallelograms. In order to provide a challenge, users may berequested to construct a parallelogram of, say, 12 square units, 24square units, 48 square units or 72 square units using selected piecesfrom the puzzle of the present invention, the units of the square beingthose depicted in FIG. 1. In similar fashion, shapes such as hexagon andparallelogram may also be set as challenges for solution to uses of thepieces of the puzzle of the present invention.

Although it is one of the features of the present invention to providepuzzles the solutions of which comprise 3 of the 4 different colours,such a feature can be presented to uses as a challenge. It has beenfound that there are several solutions providing such a feature usingthe pieces of the puzzle of the present invention.

Additionally, if it is so desired, the colours can be used to createaesthetic or artistic patterns from the pieces of the set. In additionto the above, it is believed by the inventor that the use of the puzzlepieces of the present invention enhances the learning experience ofthose who would otherwise find the learning of mathematics in generaland geometry in particular uninteresting. It is believed by theinventors that the use of the invention and the teaching of principlesinvolved may stimulate creativity in individuals. Because of theprovision of a tactile experience in respect of the demonstration ofsuch geometric principles as those found for example in Euclid'sElements Book 1, both retention and understanding of geometricprinciples may be enhanced by users of the puzzle of the presentinvention.

Although the invention has been described with reference to one specificexample, it will be appreciated by those skilled in the art that theinvention may be embodied in other forms within the broad scope andambit of the invention as herein set forth and defined by the followingclaims.

1. A dissection puzzle comprising a set of fourteen pieces, each piecein the set having an obverse face and a reverse face substantiallyparallel to the obverse face and a plurality of straight edges aboutsaid faces, and wherein said set of pieces may be formed into anassembled puzzle of straight-sided geometric shape by arranging thepieces in edge-to-edge abutting relationship, the form of the piecesbeing selected such that the area of each piece is an integer fractionof the area of the assembled puzzle, said integer fractions beingselected from a set of integer fractions, each different one from theother, at least one of said pieces having an area different from thearea of the other pieces and characterised in that: the obverse face andthe reverse face of each piece include markings selected from apredetermined set of markings such that when arranged to form theassembled puzzle and viewed from one side, each edge of each pieceadjacent another piece is adjacent a piece having a different marking.2. A dissection puzzle according to claim 1, wherein the markings arearranged such that when the pieces are arranged to form the assembledpuzzle, there is provided an alternative arrangement of pieces such thatthe assembled puzzle is comprised of a plurality of regions, one foreach marking and of equal area to each of the other regions for eachother marking.
 3. A dissection puzzle comprising a set of fourteenpieces, each piece in the set having an obverse face and a reverse faceand a plurality of straight edges about said faces, and wherein said setof pieces may be formed into an assembled puzzle of straight-sidedgeometric shape by arranging the pieces in edge-to-edge abuttingrelationship, the form of the pieces being selected such that the areaof each piece is an integer fraction of the area of the assembledpuzzle, said integer fractions being selected from a set of integerfractions each different one from the other, at least one of said pieceshaving an area different from the area of the other pieces andcharacterised in that: the obverse face and the reverse face of eachpiece include markings selected from a predetermined set of markingssuch that when arranged to form the assembled puzzle and viewed from oneside, each edge of each piece adjacent another piece is adjacent a piecehaving a different marking, and the markings are also arranged such thatwhen arranged to form the assembled puzzle, there is provided analternative arrangement of pieces such that the assembled puzzle iscomprised of a plurality of regions, one for each marking and of equalarea to each of the other regions for each other marking.
 4. Adissection puzzle comprising a set of fourteen pieces, each playingpiece in the set having an obverse face and a reverse face and aplurality of straight edges about said faces, and wherein said set ofpieces may be formed into an assembled puzzle of straight-sidedgeometric shape by arranging the pieces in edge-to-edge abuttingrelationship, the form of the pieces being selected such that the areaof each piece is an integer fraction of the area of the assembledpuzzle, said integer fractions being selected from a set of integerfractions each different one from the other, at least one of said pieceshaving an area different from the area of the other pieces andcharacterised in that: the obverse face and the reverse face of eachpiece include markings selected from a predetermined set of markingssuch that the pieces may be assembled to form an assembled puzzlecomprising contiguous areas for each marking in exact ratio for thenumber of different markings.
 5. A dissection puzzle comprising a set offourteen pieces, each playing piece in the set having an obverse faceand a reverse face and a plurality of straight edges about said faces,and wherein said set of pieces may be formed into an assembled puzzle ofstraight-sided geometric shape by arranging the pieces in edge-to-edgeabutting relationship, the form of the pieces being selected such thatthe area of each piece is an integer fraction of the area of theassembled puzzle, said integer fractions being selected from a set ofinteger fractions each different one from the other, at least one ofsaid pieces having an area different from the area of the other piecesand characterised in that: the obverse face and the reverse face of eachpiece include markings selected from a predetermined set of markingssuch that said assembled puzzle may be formed from all of the pieceswith all but one of the markings common to one side of the assembledpuzzle.
 6. A dissection puzzle comprising a set of fourteen piecesconforming to the loculus of Archimedes, each piece in the set having anobverse face and a reverse face and a plurality of straight edges aboutsaid faces, and wherein said set of pieces may be formed into anassembled puzzle of straight-sided geometric shape by arranging thepieces in edge-to-edge abutting relationship and characterised in that:the obverse face and the reverse face of each piece include markingsselected from a predetermined set of markings such that when arranged toform the assembled puzzle and viewed from one side, each edge of eachpiece adjacent another piece is adjacent a piece having a differentmarking.